Differentiation Schemes
Goal
Provide a reliable workflow to select a differentiation scheme, generate stencils, and assess accuracy for simulation discretization.
Requirements
- •Python 3.8+
- •NumPy (for stencil computations)
- •No heavy dependencies
Inputs to Gather
| Input | Description | Example |
|---|---|---|
| Derivative order | First, second, etc. | 1 or 2 |
| Target accuracy | Order of truncation error | 2 or 4 |
| Grid type | Uniform, nonuniform | uniform |
| Boundary type | Periodic, Dirichlet, Neumann | periodic |
| Smoothness | Smooth or discontinuous | smooth |
Decision Guidance
Scheme Selection Flowchart
code
Is the field smooth?
├── YES → Is domain periodic?
│ ├── YES → Use central differences or spectral
│ └── NO → Use central interior + one-sided at boundaries
└── NO → Are there shocks/discontinuities?
├── YES → Use upwind, TVD, or WENO
└── NO → Use central with limiters
Quick Reference
| Situation | Recommended Scheme |
|---|---|
| Smooth, periodic | Central, spectral |
| Smooth, bounded | Central + one-sided BCs |
| Advection-dominated | Upwind |
| Shocks/fronts | TVD, WENO |
| High accuracy needed | Compact (Padé), spectral |
Script Outputs (JSON Fields)
| Script | Key Outputs |
|---|---|
scripts/stencil_generator.py | offsets, coefficients, order, accuracy |
scripts/scheme_selector.py | recommended, alternatives, notes |
scripts/truncation_error.py | error_scale, order, notes |
Workflow
- •Identify requirements - derivative order, accuracy, smoothness
- •Select scheme - Run
scripts/scheme_selector.py - •Generate stencils - Run
scripts/stencil_generator.py - •Estimate error - Run
scripts/truncation_error.py - •Validate - Test with manufactured solutions or grid refinement
Conversational Workflow Example
User: I need to discretize a second derivative for a diffusion equation on a uniform grid. I want 4th-order accuracy.
Agent workflow:
- •Select appropriate scheme:
bash
python3 scripts/scheme_selector.py --smooth --periodic --order 2 --accuracy 4 --json
- •Generate the stencil:
bash
python3 scripts/stencil_generator.py --order 2 --accuracy 4 --scheme central --json
- •Result: 5-point stencil with coefficients
[-1/12, 4/3, -5/2, 4/3, -1/12]/ dx².
Pre-Discretization Checklist
- • Confirm derivative order and target accuracy
- • Choose scheme appropriate to smoothness and boundaries
- • Generate and inspect stencils at boundaries
- • Estimate truncation error vs physics scales
- • Verify with grid refinement study
CLI Examples
bash
# Select scheme for smooth periodic problem python3 scripts/scheme_selector.py --smooth --periodic --order 1 --accuracy 4 --json # Generate central difference stencil for first derivative python3 scripts/stencil_generator.py --order 1 --accuracy 2 --scheme central --json # Generate 4th-order second derivative stencil python3 scripts/stencil_generator.py --order 2 --accuracy 4 --scheme central --json # Estimate truncation error python3 scripts/truncation_error.py --dx 0.01 --order 2 --accuracy 2 --scale 1.0 --json
Error Handling
| Error | Cause | Resolution |
|---|---|---|
order must be positive | Invalid derivative order | Use 1, 2, 3, ... |
accuracy must be even for central | Odd accuracy requested | Use 2, 4, 6, ... |
Unknown scheme | Invalid scheme type | Use central, upwind, compact |
Interpretation Guidance
Stencil Properties
| Property | Meaning |
|---|---|
| Symmetric offsets | Central scheme (no directional bias) |
| Asymmetric offsets | One-sided or upwind scheme |
| More points | Higher accuracy but wider stencil |
Truncation Error Scaling
| Accuracy Order | Error Scales As | Refinement Factor |
|---|---|---|
| 2nd order | O(dx²) | 2× refinement → 4× error reduction |
| 4th order | O(dx⁴) | 2× refinement → 16× error reduction |
| 6th order | O(dx⁶) | 2× refinement → 64× error reduction |
Common Stencils
| Derivative | Accuracy | Points | Coefficients (× 1/dx or 1/dx²) |
|---|---|---|---|
| 1st | 2 | 3 | [-1/2, 0, 1/2] |
| 1st | 4 | 5 | [1/12, -2/3, 0, 2/3, -1/12] |
| 2nd | 2 | 3 | [1, -2, 1] |
| 2nd | 4 | 5 | [-1/12, 4/3, -5/2, 4/3, -1/12] |
Limitations
- •Boundary handling: Stencil generator provides interior stencils; boundaries need special treatment
- •Nonuniform grids: Standard stencils assume uniform spacing
- •Spectral: Not covered by stencil generator
References
- •
references/stencil_catalog.md- Common stencils - •
references/boundary_handling.md- One-sided schemes - •
references/scheme_selection.md- FD/FV/spectral comparison - •
references/error_guidance.md- Truncation error scaling
Version History
- •v1.1.0 (2024-12-24): Enhanced documentation, decision guidance, examples
- •v1.0.0: Initial release with 3 differentiation scripts